It was a controversial decision to offer the ex-Manchester City teenager a short-term deal after leaving prison, where he served eight months for causing death by careless driving.
The big defender made his debut and Kavanagh said: "He is a lad with big potential. We've tried to keep him under the radar. It was a very impressive performance given all that's gone on in his life.
"We wanted to see how he would cope and it was a huge risk and gamble which paid off."
Kavanagh heaped praise on his team, adding: "I told them to go out and make it a proper cup tie and scrap and battle for everything and they did.
"It's lovely to be in the hat for the third round. This should spur us on to better things.
"It was a terrific performance and the players showed great strength and energy and there was a huge response after the equaliser."
It was the Bees' first defeat in 10 games and came just hours after losing Rosler to Wigan.
James Berrett's first-half goal for Carlisle was cancelled out by a Pascal Chimbonda own goal after half-time, but two Lee Miller strikes, one a penalty, helped put the Cumbrians into the draw for the third round before Farid El Alagui kept Bees' hopes alive with his late header.
Berrett scored a 20-yarder to give Carlisle a half-time lead. However the Cumbrians were caught out by Alan McCormack's hard, low cross from the right, which led to Chimbonda diverting the ball into the net from six yards out to give Bees their 63rd-minute equaliser
Carlisle restored their lead in the 73rd minute when Miller received the ball from Berrett and from 12 yards found the bottom corner with his low first-time shot.
Miller made it 3-1 in the 77th minute from the penalty spot after McCormack had fouled David Amoo. Miller made no mistake, blasting the kick low to Lee's right.
Carlisle had some nervous moments after El Alagui headed home from Sam Saunders' set piece late on, but saw the game out.